ON THE MOTION OF PENDULUMS. 101 respectively, and that, whether the time t be great or small. Hence if we subtract the logarithm of the second factor from that of the first we shall get the logarithm of the factor due to the action of the first force alone. But if we put each factor under the form 1 — m, and subtract the m of the second factor from the m of the first, we shall not get the m due to the first force alone, unless t be small enough to allow of our neglecting the squares of ct and c't, or at least the product ct. c't In truth, when t = r, the quantities m are sufficiently small to be treated in Coulomb's manner without any material error, since the corrected values of log (1 — m), obtained in the two ways, would only differ in the 4th place of decimals. The numbers given in the last column of the above table were calculated from the formula (164), on substituting for log(l — m)""1 the numbers found in the first three lines of the 4th column, corrected by subtracting 0*0058. The mean of the three results is 0-05557, but the three experiments are not equally valuable for the determination of \///. For the three numbers from which yX was deduced are 0*0510, 0*0152, 0*0077, and a given error in the first of these numbers would produce a smaller error in vV than that which would be produced by the same error in the second, still more, than that which would be produced by the same error in the third. If we multiply the three values of vX by 510, 152, and 77, respectively, and divide the sum of the products by 510 +152 H- 77 or 739, we get O'Ooool. We may then take 0*555 as the result of the experiments. Assuming v/X = 0*0555 we have log (1 - m)~l from experiment 0-0568 in No. 1, ............... from theory 0-0571 0-021 in No. 2, 0-0206 0-0135 in No. 3, 0-0137 difference -0-0003 + 0-0004 -0-0002 65. So far the accordance of the theoretical and observed results is no very searching test of the truth of the theory. For, in fact, the theory is involved iu the result only so far as this, that it shews that the resistance experienced by a given small element of a disk oscillating in a given period varies as the linear velocity; since the difference of periods in Coulomb's experiments was so small that the effects thence arising would be mixed up with errors of observation. This law is so simple that it might 13 I